For the reader's convenience, we reproduce the proof of the above result in Appendix  and compare some of the steps with the proof of Theorem REF . Observe that the bound (REF ) is closely related to the upper bound (REF ) that holds for unconstrained least squares in the well-specified setup with Gaussian design: both bounds do not depend on the magnitude of the covariates, specific properties of the covariance structure, and the norm of the optimal linear predictor \(\omega ^* = \operatornamewithlimits{arg\,inf}_{\omega \in \mathbb {R}^d}R(\omega )\) . The difference is that \(R(\omega ^*)\)  is replaced by \(m^2\) . It is reported in [1]} that the authors could not prove a bound similar to (REF ) for the least squares estimator. Given the lower bound (REF ) and Proposition REF , it is not surprising. In our setting, the performance of least squares is affected by the boundedness constants \(r, b\) , which can be arbitrarily bad in the theorem above.
